**Volume 10, Issue 11, pp 5608--5621**

**Publication Date**: 2017-11-10

**Cuimei Jiang**
- School of Science, Qilu University of Technology, Jinan, Shandong 250353, P. R. China

**Fangfang Zhang**
- School of Electrical Engineering and Automation, Qilu University of Technology, Jinan, Shandong 250353, P. R. China

**Haiyong Qin**
- School of Mathematics, Qilu Normal University, Jinan, Shandong 250013, P. R. China

**Tongxing Li**
- School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, P. R. China

This paper is concerned with adaptive control for anti-synchronization of a class of uncertain fractional-order chaotic complex systems described by a unified mathematical expression. By utilizing the recently established result for the Caputo fractional derivative of a quadratic function and employing the adaptive control technique, we design controllers and some fractional-order parameter update laws to anti-synchronize two fractional-order chaotic complex systems with unknown parameters. The proposed method has generality, simplicity, and feasibility. Moreover, anti-synchronization between uncertain fractional-order complex Lorenz system and fractional-order complex Lu system is implemented as an example to demonstrate the effectiveness and feasibility of the proposed scheme.

Adaptive control, anti-synchronization, fractional-order chaotic complex system, quadratic Lyapunov function

[1] S. K. Agrawal, S. Das, Projective synchronization between different fractional-order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique, Math. Methods Appl. Sci., 37 (2014), 2164–2176.

[2] N. Aguila-Camacho, M. A. Duarte-Mermoud, J. A. Gallegos, Lyapunov functions for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2951–2957.

[3] D. Baleanu, G.-C. Wu, Y.-R. Bai, F.-L. Chen, Stability analysis of Caputo-like discrete fractional systems, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 520–530.

[4] S. Bhalekar, V. Daftardar-Gejji, Synchronization of different fractional order chaotic systems using active control, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3536–3546.

[5] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. Int., 13 (1967), 529–539.

[6] L.-P. Chen, Y. Chai, R.-C. Wu, Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems, Phys. Lett. A, 375 (2011), 2099–2110.

[7] J.-Y. Chen, C.-D. Li, T.-W. Huang, X.-J. Yang, Global stabilization of memristor-based fractional-order neural networks with delay via output-feedback control, Modern Phys. Lett. B, 2017 (2017), 19 pages.

[8] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approch for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3–22.

[9] A. M. A. El-Sayed, E. Ahmed, H. A. A. El-Saka, Dynamic properties of the fractional-order logistic equation of complex variables, Abstr. Appl. Anal., 2012 (2012), 12 pages.

[10] X. Gao, J.-B. Yu, Chaos in the fractional order periodically forced complex Duffing’s oscillators, Chaos Solitons Fractals, 24 (2005), 1097–1104.

[11] A. K. Golmankhaneh, R. Arefi, D. Baleanu, Synchronization in a nonidentical fractional order of a proposed modified system, J. Vib. Control, 21 (2015), 1154–1161.

[12] A. Jajarmi, M. Hajipour, D. Baleanu, New aspects of the adaptive synchronization and hyperchaos suppression of a financial model, Chaos Solitons Fractals, 99 (2017), 285–296.

[13] C.-M. Jiang, S.-T. Liu, C. Luo, A new fractional-order chaotic complex system and its antisynchronization, Abstr. Appl. Anal., 2014 (2014), 12 pages.

[14] C.-M. Jiang, S.-T. Liu, D. Wang, Generalized combination complex synchronization for fractional-order chaotic complex systems, Entropy, 17 (2015), 5199–5217.

[15] A. Kiani-B, K. Fallahi, N. Pariz, H. Leung, A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 863–879.

[16] C.-G. Li, G.-R. Chen, Chaos and hyperchaos in the fractional-order R¨ossler equations, Phys. A, 341 (2004), 55–61.

[17] C.-G. Li, G.-R. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals, 22 (2004), 549–554.

[18] H.-L. Li, C. Hu, Y.-L. Jiang, Z.-L. Wang, Z.-D. Teng, Pinning adaptive and impulsive synchronization of fractional-order complex dynamical networks, Chaos Solitons Fractals, 92 (2016), 142–149.

[19] J. Liu, Complex modified hybrid projective synchronization of different dimensional fractional-order complex chaos and real hyper-chaos, Entropy, 16 (2014), 6195–6211.

[20] X.-J. Liu, L. Hong, L.-X. Yang, Fractional-order complex T system: bifurcations, chaos control, and synchronization, Nonlinear Dynam., 75 (2014), 589–602.

[21] J. Liu, S.-T. Liu, W. Li, Complex modified generalized projective synchronization of fractional-order complex chaos and real chaos, Adv. Difference Equ., 2015 (2015), 16 pages.

[22] S. Liu, X. Wu, X.-F. Zhou, W. Jiang, Asymptotical stability of Riemann–Liouville fractional nonlinear systems, Nonlinear Dynam., 86 (2016), 65–71.

[23] S. Liu, X.-F. Zhou, X.-Y. Li, W. Jiang, Asymptotical stability of Riemann–Liouville fractional singular systems with multiple time-varying delays, Appl. Math. Lett., 65 (2017), 32–39.

[24] J.-H. Luo, H. Liu, J.-F. Yang, Robust synchronization of uncertain fractional order chaotic systems, IEICE Trans. Fundamentals, E98-A (2015), 2109–2116.

[25] C. Luo, X.-Y. Wang, Chaos generated from the fractional-order complex Chen system and its application to digital secure communication, Int. J. Modern Phys. C, 2013 (2013), 23 pages.

[26] C. Luo, X.-Y. Wang, Chaos in the fractional-order complex Lorenz system and its synchronization, Nonlinear Dynam., 71 (2013), 241–257.

[27] P. Muthukumar, P. Balasubramaniam, Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography, Nonlinear Dynam., 74 (2013), 1169–1181.

[28] Z. M. Odibat, Adaptive feedback control and synchronization of non-identical chaotic fractional order systems, Nonlinear Dynam., 60 (2010), 479–487.

[29] H.-Y. Qin, C.-H. Zhang, T.-X. Li, Y. Chen, Controllability of abstract fractional differential evolution equations with nonlocal conditions, J. Math. Computer Sci., 17 (2017), 293–300.

[30] G.-Q. Si, Z.-Y. Sun, H.-Y. Zhang, Y.-B. Zhang, Parameter estimation and topology identification of uncertain fractional order complex networks, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 5158–5171.

[31] J.-W. Sun, W. Deng, G.-Z. Cui, Y.-F. Song, Real combination synchronization of three fractional-order complex-variable chaotic systems, Optik, 127 (2016), 11460–11468.

[32] H. Targhvafard, G. H. Enjace, Phase and anti-phase synchronization of fractional-order chaotic systems via active control, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4079–4088.

[33] Z. Wang, X. Huang, N. Li, X.-N. Song, Image encryption based on a delayed fractional-order chaotic logistic system, Chin. Phys. B, 21 (2012), 6 pages.

[34] Z. Wang, X. Huang, G.-D. Shi, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Comput. Math. Appl., 62 (2011), 1531–1539.

[35] G.-C. Wu, D. Baleanu, Chaos synchronization of the discrete fractional logistic map, Signal Process., 102 (2014), 96–99.

[36] G.-C. Wu, D. Baleanu, W.-H. Luo, Lyapunov functions for Riemann–Liouville-like fractional difference equations, Appl. Math. Comput., 314 (2017), 228–236.

[37] G.-C. Wu, D. Baleanu, H.-P. Xie, F.-L. Chen, Chaos synchronization of fractional chaotic maps based on the stability condition, Phys. A, 460 (2016), 374–383.

[38] X.-J. Wu, D.-R. Lai, H.-T. Lu, Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes, Nonlinear Dynam., 69 (2012), 667–683.

[39] H.-Q. Wu, X.-X. Zhang, S.-H. Xue, L.-F. Wang, Y. Wang, LMI conditions to global Mittag–Leffler stability of fractionalorder neural networks with impulses, Neurocomputing, 193 (2016), 148–154.

[40] L.-X. Yang, J. Jiang, Complex dynamical behavior and modified projective synchronization in fractional-order hyper-chaotic complex Lü system, Chaos Solitons Fractals, 78 (2015), 267–276.

[41] C. Yin, S. Dadras, S.-M. Zhong, Y.-Q. Chen, Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach, Appl. Math. Model., 37 (2013), 2469–2483.

[42] Y.-P. Zhang, J.-T. Sun, Chaotic synchronization and anti-synchronization based on suitable separation, Phys. Lett. A, 330 (2004), 442–447.